Question

How do you divide `(x^4 + 2x^3 +3x -1 )/ (x^2 + 2)`?

Answer 1

`x^4+2x^3+3x-1=x^2+2x-2-(x-3)/(x^2+2)`

Explanation:

`x^4+2x^3+3x-1`

= `(x^2xx x^2)+ul(2x^2)+2x^3-ul(2x^2)+3x-1`

= `x^2(x^2+2)+x^2xx2x+ul(2x xx2)-2x^2-ul(4x)+3x-1`

= `x^2(x^2+2)+2x(x^2+2)-2(x^2+2)-4x+3x+4-1`

= `(x^2+2x-2)(x^2+2)-x+3`

Hence, `(x^4+2x^3+3x-1)/(x^2+2)`

= `x^2+2x-2-(x-3)/(x^2+2)`

Answer 2

`x^2+2x-2-(x-3)/(x^2+2)`

Explanation:

Using place keepers such as `0x^2` to make alignment and calculations more straightforward.

`" "x^4+2x^3+0x^2+3x-1`
`color(magenta)(x^2)(x^2+2) ->ul(x^4+0x^3+2x^2) larr" Subtract`
`" "0 +2x^3-2x^2+3x-1`
`color(magenta)(2x)(x^2+2) ->" "ul(2x^3+0x^2+4x ) larr" Subtract"`
`" "0 -2x^2-x-1`
`color(magenta)(-2)(x^2+2)->" "ul(-2x^2+0x-4 ) larr" Subtract"`
`color(magenta)(" "0-x+3 larr" Remainder")`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(magenta)(x^2+2x-2 +[(-x+3)/(x^2+2)])`

`x^2+2x-2-(x-3)/(x^2+2)`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
If you are not sure about the change in sign for the bracket consider this:

Multiply by (+1) but in the form of `(-1)/(-1)`

`(-1)/(-1)[(-x+3)/(x^2+2)]`

`(-1)[(-x+3)/(-1) xx1/(x^2+2)]`

`(-1)[(+x-3) xx1/(x^2+2)]`

`-[(x-3)/(x^2+2)]" "=" "-(x-3)/(x^2+2)`